Straight Lines Class 11 | Ch-09

Straight Lines Class 11 Chapter 09

Basic concepts of straight line chapter 9 standard XI, slope of the line, different forms of equation of lines and general equation of line.

To understand this topic properly we should know the coordinate geometry. Coordinate geometry is the field of mathematics in which we solve the geometrical problems with the help of algebra. In coordinate geometry class 10 we learnt about distance formula, section formula, mid point formula and area of triangle.

Angle of inclination of the line

The angle θ  made by the line l with positive direction of x- axis and measured anti-clockwise is called the angle of inclination of the line

Angle of inclination of the x-axis is 0^o.  Angle of inclination of all the lines parallel to the x- axis is also 0^o.

Angle of inclination of the y-axis is 90^o.  Angle of inclination of all the lines parallel to the y- axis is also 90^o.

Slope of the line

If  'θ' is the angle of inclination of the line l then tanθ is called the slope or gradient of the line l. Slope is denoted by m

Therefore slope of line l = m = tanθ

Slope of x-axis = tan0 = 0

Slope of y - axis = tan90^o = ∞  or undefined

Slope of the line passing through the two points P(x₁, y₁), and Q(x₂, y₂₎

   \[m=tan\theta =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]Two lines are parallel if their slopes are equal or  m₁ = m₂  

Two lines are perpendicular if product of their slopes equal to -1 or  m₁ x m₂ = -1

 

Angle between two lines

If  m₁ , m₂  are the slopes of two given lines and if   'θ' is the angle between two lines then  \[tan\theta =\left |\frac{m_{1}-m_{2}}{1+m_{1}m_{2}} \right |=\pm \frac{m_{1}-m_{2}}{1+m_{1}m_{2}}\]

Collinearity of three points

Three points A, B, C are collinear to each other if  

Slope of AB = Slope of BC

Different forms of equation of line

We know that every point in a plane have infinitely many points lie on it. If (x, y) is any arbitrary point on it then it represent every point on the line.

Equations of Horizontal Lines 

If a horizontal line L₁ or L₂  is at a distance a from the x-axis then ordinate of every point lying on the line is either a or –a. 

Therefore equation of the line is either of the form y = a or  y = - a

If a line is above the x-axis then choose +ve sign and if a line below the x-axis then choose –ve sign.

Equations of Vertical Lines 

If a vertical line L₁ or L₂ is at a distance b from the y-axis then ordinate of every point lying on the line is either b or –b. Therefore equation of the line is either of the form x = b or  x = - b

If a line is to  the right of  y-axis then choose +ve sign and if a line is to the left of  the y-axis then choose –ve sign.

Point Slope form of equation of line

This is also known as one point form of equation of line.

Let a line L with slope m passes through the point Q(x₁, y₁). If P(x, y) is any arbitrary point on the line L, Then equation of line  is given by 

   y - y₁ = m(x - x₁)  where

 \[m=tan\theta =\frac{y-y_{1}}{x-x_{1}}\]

Two Point form of equation of line

Let a line L passing through two points P(x₁, y₁) and Q(x₂, y₂). Slope m of the line is given by \[m=tan\theta =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]. If R(x, y) is any arbitrary point on the line then equation of the line is given by \[y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\]

Slope intercept form of equation of line

Let a line L with slope m cut the y-axis at a distance c from the origin at point (0, c). Then  c is called the y-intercept of the line. If P(x, y) is any arbitrary point on the line then equation of the line is given by

y - c = m(x - 0)      ⇒   y = mx + c

If line L with slope m cut the x-axis at a distance c from the origin at point            (c, 0). Then  c is called the x-intercept of the line. If P(x, y) is any arbitrary point on the line then equation of the line is given by

y - 0 = m(x - c)      ⇒   y = m(x + c)

Intercept form of equation of line

Let L is a line which make  intercept a on the x-axis and intercept b on y-axis, then slope of the line is given by \[Slope(m)=\frac{b-0}{0-a}=\frac{-b}{a}\]Equation of line is given by\[y-0=\frac{-b}{a}(x-a)\]\[ay=-bx+ab\]\[ay+bx=ab\]\[\frac{ay}{ab}+\frac{bx}{ab}=\frac{ab}{ab}\]\[\frac{y}{b}+\frac{x}{a}=1\]\[\frac{x}{a}+\frac{y}{b}=1\]This equation is called the intercept form of equation of line.

Normal form of equation of line

Let L is a line whose perpendicular distance from the origin is p and perpendicular OA makes an angle θ with the x-axis measured anti-clockwise direction, then Normal form of the equation of line is given by

xcosθ + ysinθ = p 

General equation of line

An equation of the form  ax + by + c = 0 is called linear equation or general equation of line. In this equation a and b both collectively cannot be zero.

To find the slope from the general eqn. of line\[Slope(m) = \frac{-coefficient of x}{coefficient of y}=\frac{-a}{b}\]

Perpendicular distance of a point from the line

Perpendicular  distance  of  point  (X₁, Y₁)  from   the   line   ax + by + c = 0  is given by\[d=\left | \frac{ax_{1}+by_{1}+c}{\sqrt{a^{2}+b^{2}}} \right |\]

Distance between two parallel lines

Let two parallel lines are :  

ax + by + c₁ = 0,

ax + by + c₂ = 0

Perpendicular distance between these two parallel lines is given by

\[d=\left | \frac{c_{1}-c_{2}}{\sqrt{a^{2}+b^{2}}} \right |\]

Method of converting the general equations into the Normal form of equation of line.

Let us suppose that the equation of the line in general form is 

\[x-\sqrt{3}y+8=0\]

Bring the constant term to the right hand side and keep in mind that constant term should be positive.

\[-x+\sqrt{3}y=8\]

\[Dividing\; on\; both\; sides\; by\; \; \sqrt{a^{2}+b^{2}} \; or\; \; \sqrt{(-1)^{2}+\left (\sqrt{3} \right )^{2}}=2\]

\[\frac{-x}{2}+\frac{\sqrt{3y}}{2}=\frac{8}{2}\]

\[\frac{-x}{2}+\frac{\sqrt{3y}}{2}=4\]

\[Let \; \; cos\theta = -\frac{1}{2}\; and\; \; sin\theta =\frac{\sqrt{3}}{2}\; and\; p\; =4\]

\[tan\theta =\frac{sin\theta }{cos\theta }=\frac{\sqrt{3}/2}{-1/2}=-\sqrt{3}\]

\[tan\theta =-\sqrt{3}=-tan\frac{\pi }{3}=tan\left ( \pi -\frac{\pi }{3} \right )=tan\left ( \frac{2\pi }{3} \right )\]

\[\Rightarrow \theta =\frac{2\pi }{3}\; or\; 120^{o}\]

Normal form of equation of line is \[xcos\theta +ysin\theta =p\]

\[xcos\left (\frac{2\pi }{3} \right ) +ysin\left (\frac{2\pi }{3} \right ) =4\]

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